Electronic equation solver



G. w. BROWN ET Al. 2,454,549

ELECTRNIC EQUATION SOLVER Nov. 23, 1948.

Filed Aug. 16. 1946 7 Sheets-Sheet 1 A'eA W Azz l.

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u 4INVENTORE cinesi m eow/v a ATTORNEY Nov. 23, 1948. G. w. BROWN ET Al-ELECTRONIC EQUATION SOLVER Illll 1` f 1 INVENTORS Genese w eow/v & 50W/NA. ofe@ y w TORNEY Filed Aug. 16, 1946 Nov. 23, 1948. G. w. BROWN ET ALELECTRONIC EQUATION SOLVER 7 Sheets-Sheet 5 Filed Aug. 16, 1946 Nov. 23,1948. G. w. BROWN ET Al- ELEGTRONIG EQUA'IION SOLVER 7 Sheets-SheetFiled Aug. 16, 1946 @BEB IML... A ORNEY NOV. 23, 1948. G, wl BROWN ET ALELECTRONIC EQUATION soLvER 7 Sheets-Sheet 5 Filed Aug. 16. 1946 aus .wwms. xnwwwk NOV- 23,` 1948. G. w. BRowN ET A; 2,454,549

ELECTRONIC EQUATION SOLVER Filed Aug. .16, 194e f 7 sheets-sheet 6VENTORS ow/N A. @maize ATTO RNEY N9V- 23 1948- G. w. BROWN ET Al.2,454,549

ELECTRONIC EQU'JIONl SOLVER Filed Aug. 1e, 194e Y 'I sheets-sheet 7/ajagaao `Patented Nov. 23, 1948 vUNITED STATES PATENT GFFICE ELECTRONICEQUATION SOLVER George W. Brown, Granbury, and Edwin A. Goldberg,Princeton, N. J., assignors to Radio Corporation of America., acorporation of Delaware Application August 16, 1946, Serial No. 690,865

3 Claims. l

This invention relates to a method of and means for electronicallysolving linear simultaneous equations. As -is well known, simultaneousequations are two or more equations which are satisfied by the same setsof values of the unknown quantities. The solution of such equations byconventional methods becomes more and more involved as the number ofequationsv and unknowns increases. Conventional methods areunsatisfactory for the solution of systems of simultaneous equationsinvolving, say, ve to ten equations and unknowns because the process istoo tedious, particularly when a solution is attempted by trial anderror methods. It is therefore the primary object of this invention toprovide a new method of and means for electronically solvingsimultaneous equations.

The electronic solution of simultaneous equations in accordance withthis invention utilizes the amplitude and phase of a plurality ofvoltages to represent the value and sign of the known terms of thevarious equations, and the method is therefore a continuous one asdistinguished from the 'step or counter methods of calculation whichhave sometimes been used for electronic computation. It will be shownthat by establishing a first series of voltages representative of th'evalues ofthe constant terms of a plurality of simultaneous equations,establishing a lsecond series of voltages representative of thecoefficients of the unknown terms in each equation, and properlycombining with each of the iirst series of voltages the ones of thesecond series which represent terms in a given equation, the resultantvoltages are indicative of the unknown terms.

In carrying out this invention a number of amplifiers, equal to thenumber of equations to be solved, are interconnected by mutual feedbackcircuits in such a manner that when input voltages are adjusted tocorrespond in amplitude and phase to the value and sign of the constantterms of the equations and are applied to the respective amplifiers, andfeedback voltages are selected which are proportional to predeterminedfractions of the output of each amplifier, and are fed back into theinput of each amplifier, respectively, and where the predeterminedfractional values are determined by the coefficients of thecorresponding unknowns, then the amplitudes and phases of the outputvoltages of the ampliers determine the values and signs of the unknowns.It is therefore a further object of this invention to provide anelectronic equation solver capable of solving a plurality ofsimultaneous equations.

More particularly, the system which illustrates a preferred embodimentof this invention and which may be utilized in practicing the methodherein described, employs a number of voltage dividers of thepotentiometer type, each having an accurately calibrated dial scaleadjustable to three pertinent places, for determining the input voltagesin terms' of a 'selectable percent of a reference voltage and also forderiving from each amplifier output the desired fractional feedbackvoltages. A system for solving up to ten equations including ten unknownvalues or roots, would have, foreach equation, one dial for setting theconstantterm potentiometer plus ten dials for setting the potentiometerswhich provide the ten coeicients of the ten unknowns, or a total ofeleven dials and potentiometers per equation. For the ten equationsthere would be a total of dials and potentiometers, preferably arrangedin ten horizontal rows of eleven dials eachl Of course the solution ofsay, five equations, would only involve the use of the six dials of anyve rows. It is therefore a further object of this invention to provide asimultaneous equation solver in which the known values may be applied tothe device by setting at the appropriate values a number of calibrateddials so that the device is direct reading.

The novel features that are considered characteristic of this inventionare set forth with particularity in the appended claims. The inventionitself, however, both as to its organization and method of operation, aswell as additional objects and advantages thereof, will best beunderstood from the following description when read in connection withthe accompanying drawings,

in which Figure 1 is a schematic diagram illustrating the principle ofoperation of this invention;

Figure 2 is a schematic diagram illustrating the method of connectingthe components of a preferred embodiment of an equation solver for thesolution of n simultaneous equations;

Figure 3 is an equivalent diagram of a feedback amplifier useful inexplaining the theory of operation of the interconnected feedbackampi-ier networks;

Figure 4 is the circuit diagram of a 'stable amplifier suitable for usein the equation solver;

Figure 5 is the circuit diagram of a high pass filter utilized in theequation solver;

Figure 6 is the circuit diagram of a low pass filter utilized in theequation solver;

Figure 'i is the circuit diagram of a band pass filter utilized in theequation solver;

. Aii--fraction Figures 8 and 9 are graphs illustrating the phase shiftcharacteristics of the amplifier shown in Fig. 4;

Figure is a schematic diagram of the signal generator, bridge andindicator components of the equation solver; and

Figures 11 and 12 are graphs illustrating the attenuationcharacteristics of the filters included in the amplifier of Fig. 4.

Fig. 1 illustrates in simplied form (for n=2) the theory of operation ofa device for solving a system of n simultaneous equations of the typeY1=A11X1+Ai2X2 AinXn (1) Y2=A21Xi|A22X2 AznXn (2) and Y1i=An1Xl+An2XZAmiXn (3) where X1,X2 Xn are the unknowns; A11, A21 A111 the constantcoeflicients of X1; A12, A22 Anz the constant coeilicients of X2,

etc.; and Y1, Y2 Y" are constants.

The basic system involves the application of alternating input voltagesto a number of ampliners, the number being the same as the number ofsimultaneous equations to be solved, which are so interconnected byresistance networks that the output voltages are proportional to thevalues of the corresponding unknowns.

Thus, in Fig. l, a signal generator I produces an alternating referencevoltage having a frequency of the order of 1000 c. p. s. Its output isconnected to terminals 2, 3 across which two potentiometers 5 and 1 areconnected. The movable arms are connected, respectively, to the X1 andX2 amplifiers 9 and II through summing resistors I3 and I5. One inputterminal of each amplifier is grounded.

The characteristics and details of the amplifier will be describedhereinafter. For the present assume the gain of each amplifier has thevalue n. The output of each amplifier is applied to two potentiometers,I1 and I9 being connected to amplifier X1 while 2I and 22 are connectedto amplifier X2. The output of potentiometer I1 is applied to the inputof its associated amplifier 9 through a summing resistor 25 while theoutput of potentiometer I9 is applied to the input of the otheramplifier II through a summing resistor 21. Similarly, the output ofpotentiometer 2| is applied to amplifier II and that of potentiometer 23to the input of amplier 9 through summing resistors 29 and 3|,respectively. The value R of each of the summing resistors should behigh with respect to the total potentiometer resistance of 1000 ohms,and may be, for eX- ample, of the order of 500,000 ohms.

Considering the steady state operating conditions, let the various termsbe as follows:

X1=output voltage of amplier X1 X2=output voltage of amplifier X2 of X1derived from potentiometer I1 Az1=fraction eter I9 Azz=fraction eter 2IA12=fraction eter 23 Y1=voltage input derived from potentiometer 5Y2=voltage input derived from potentiometer 1 E1=voltage across input toamplifier X1 E2=voltage across input to ampliiier X:

of X1 derived from potentiomof X2 derived from potentiomof X2 derivedfrom potentiom- Transposing and substituting the value of E from (4) and(5) -Y1= (A11-shi) X1+A12X2 (8) and If n at the operating frequency isknown the term 1l/n may be subtracted from the terms A11 and A22.Equations 8 and 9 are then seen to be the same as 1 and 2 except thatthe sign of Y is reversed. Full equivalence can easily be established byproperly phasing Y with respect to X. This may be done within theamplifier or by any conventional means as is well known. The gain ofmost amplifiers is subject to wide variations due to tube life, changesin line voltage and the like. While the gain could be measured as oftenas necessary, rather than measuring and substracting 3/11. from eachterm, it is preferred to make the gain n at the operating frequency solarge that the term -3/11 can be neglected, and then Equations 8 and 9will be identical with 1 and 2, the same phase adjustment being assumed.The actual value of gain depends upon the accuracy desired. A value for,u. of 15,000 for a ten equation solver has been found to besatisfactory and to produce an error within the limitations of othercircuit parameters. Thus, since the operating equation of the circuit ofFig. 1 is identical to a simultaneous pair of equations in two unknowns,it follows that if the system isA stable, and when the known values areapplied as indicated above, the

measured value of voltages X1 and X2 will sat-' isfy the unknown valuesof the simultaneous equations. That is, if the input potentiometers 5and 1 are adjusted so that the voltages Y1 and Y2 measured to ground areequal to the values of the constants Y1 and Y2 of any two simultaneousequations, of the type illustrated by Equations 1 and 2, and the outputpotentiometers are set at fractional values of the output voltages equalto the coefficients of X1 and X2, then the output voltages X1 and X2will be equal t-o the values of the unknowns which satisfy theequations. The signal generator voltage may be maintained at someconvenient standard value so that the input potentiometer dials will bedirect reading, or by bridging the X voltage against the generatorvoltage. the output indicated will be independent of the inputamplitude, as will be more fully explained below. The outputpotentiometers may also be calibrated in terms of percent of theamplifier output voltage derived at any setting of the movable contactarms, and will also be direct reading. Single phase operation has beenillustrated in the simplified showing of Fig. 1. It is to be understood,however, that push-pull driving circuits may be employed and eitherpotentlometers with grounded center taps or suitable switches utilizedto establish the proper phase orusign for the volttages representing theequation. To accomplish this phase reversal most efficiently, thepotenti ometers may be connected alternatively in the anode and cathodecircuits of the associated vacuum tubes. As is well known, this willreverse the phase of the current, and thus reverse the phase of theoutput voltage.

The accuracy of the system depends to a large vabout 3% feet.

degree upon, the accuracy with which the potentiometers can be set tothe desired resistance ratio, and therefore a potentiometer of the typeknown as a Micropot is preferred. This type of potentiometer consists ofa turn helical resistance element with an effective length of Preferablythe potentiometers are equipped with counter type dials indicating thecontactor position by a decimal system calibration which thus permitsthe equations to be set up directly with reasonable accuracy.

The circuit illustrated in Fig. 1 is suitable for solving twosimultaneous equations involving two unknowns where the voltages aremeasured with a suitable voltmeter. The system may be extended toinclude the solution of n simultaneous equations involving n unknowns byproviding n Y networks, and n amplifiers, each amplifier feeding into nparallel connected A networks. The scheme for connecting n systems isillustrated in Fig. 2, to which reference is now made.

As suggested briefly above, phase reversal is accomplished in thepreferred embodiment of this invention, by connecting the potentiometers||9 in the plate circuits of the associated driver tubes for the phaseswhich can be called and in the cathode circuits of the same tubes forthe phase. In order to insure constant positive and negative phasebalance as the Micropots are switched from one circuit to the other,fixed resistors |2| of equal value are simultaneously switched so as toreplace the Micropots. Thus the plate and cathode impedances areconstant at all times so long as the load connected to the Micropot isnegligible. In Fig. 2, the combination of a Micropot H9, or equivalentpotentiometer, the balancing resistor |2| and the polarity reversingswitch |23 shown enclosed in a dotted line, is hereinafter referred toas a network. Each such network has been given a common referencenumeral ||1. The individual networks are preferably identified by theirfunctional representation which indicates their place in the generalform of the Equation 1, 2 or 3. Thus the subscript of the Y networksindicates which of the n equations the network in question is to beassociated with. In setting up the value of the constant Y1 in Equation1, network Y1 would be used, and so forth. Two subscripts identify eachA network. The rst of these identies the equation while the secondvidentifies the unknown which it modifies.

In order to obtain the voltage representative of the constant Y, eachnetwork Y1, Y2, Y11 is connected to the plate P and cathode K terminalsof the signal generator, bridge and indicator device which is shown indetail in Fig. l0. The and J1-f connections indicated in each network||1 are to be connected to the (ground) and terminals of a source of D.C. potential of 150 volts to provide plate voltage to the driver tubesas will appear necessary from a consideration of Fig. 10.

The output of network Y1 is connected through an impedance such 'as asumming resistor |251 shunted by a phasing capacitor |211 to the inputof amplifier X1. Similarly the output of network Y2 is connected througha summing resistor |252 shunting by a phasing capacitor |212 to theinput of amplifier X2 and the output of network Y11 is connected to theinput of amplifier X11 through resistor |2511 and capacitor |2111. Toobtain phase selection in the A networks, output terminals |09 and ofthe X ampliers connect internallyvto the cathode and plate electrodes ofthe amplier output tubes, as will appear from a consideration of Fig. 4.These output terminals are connected to the polarity reresistors |291,|292,

versing switches of networks A11, A21, A111 so that when any switch |23is to the left the M- cropot is in the circuit from plate to B+ and thebalancing resistor |2| is in the circuit from cathode to B- or ground,as shown, similarly, the output of amplifier X2 connects to the polarityreversing switches of networks A11, A22, A112 and the output ofamplifier X11 is connected to networks A111, A211, A1111.

'Ihe movable contact of each Micropot of the A networks is connectedthrough a summing resistor, which may be shunted with a phasingcapacitor, to the input of the amplifier identified by the first numeralof the A network subscript.

and the associated capacitors. Similarly, the out-v put of each networkof the series A21, A22, A211 is connected to the input of amplifier X2through |2911, respectively, and the output of each network of theseries A111, A112, A1111 is connected to the input of amplifier X11through resistors |3|1, |3|2, |3|11, respectively, so that the feedbackvoltages of each amplifier are applied to the input terminals of all theamplifiers, respectively.

For convenience, the output voltages of amplifiers X1, X2, X11 aremeasured successively. To accomplish this each amplifier output isconnected to a correspondingly numbered terminal X1, X2, X11 of thedevice |20 which includes the indicator shown in detail in Fig. 10.

It is to be noted that the first equation (1) of n simultaneousequations would be set up in the equation solver by adjusting network Y1in accordance with the constant term, and the A networks in the firstcolumn A11, A12, A111 in accordance with the corresponding coefficientsof X1, X2, X11. Similarly, Equation 2 would be applied to the Y2 networkand the A networks of the second column, and so on. It is also to benoted that the A networks in any horizontal row are the coefficients ofa given unknown. This arrangement of rows and columns may be retained inthe physical layout of the network potentiometers, or the A rows andcolumns may be physically interchanged.

The construction of a practical device of the type illustrated in Figs.1 and 2 requires the solution of certain problems regarding systemstability, and accuracy. The dynamic stability and the accuracy ofsolution are intimately related to the characteristic roots of thematrix of coefdcients ofthe equations. For definition and properties ofthese terms, reference is made to the textbook of Bcher Introduction toHigher Algebra," or any other textbook of higher algebra. Briefly,associated with a square matrix of n rows and n columns, with realelements, there are associated n characteristic roots )11 which are realor occur in conjugate complex pairs. With respect to stability andaccuracy it can be Vshown that the behavior of the system of Fig. 2

complex gain M, M, in as the case may be, is constant at allfrequencies, and a summing resistor nR. Thus the analysis of the largenumber of interconnected networks of Fig. 2 may be reduced to thedetermination of the performance of a conventional equivalent where theperformance characteristics are well known. The interconnected networkwill then be stable when all n. of the independent equivalent networks,corresponding to the n possible characteristic roots, are stable, andnot otherwise.

In order that there exist at least one series of equations which may besolved with an acceptable degree of accuracy, it is at least necessarythat the X amplifier be capable of stable operation with sufficientfeedback to insure the degree of accuracy required. The operation of thefeedback amplifier of Fig. 3 may be expressed. Yl' :0, n +1 xl,

n n u u A YDI=(}\R 7L'|. 1)XI It may be seen that the degree of accuracyis determined by the ratio of )l to the value of at the operatingfrequency. Also a fairly large i makes more eiiicient use of thepotentiometer dial scale factor. Thus if the equation solver is to beused primarily in the solution of equations whose characteristic rootsare real and positive (assuming the amplified output to be 180 out ofphase with the input voltage without feedback) it is sufficient that theamplifier be capable of sustaining the feedback with relatively littlephase margin. An amplifier with phase margin of :90 will permit stablesolution of any set of equations whose characteristic roots, whilecomplex, have positive real parts.

Referring to Fig. 4, there is illustrated an amplifier having therequired characteristics which may be used in the complete equationsolver illustrated by Fig. 2. The input voltage is applied to the gridof a high gain amplifier tube 35 of the 6AK5 type, through a 2;fdcoupling capacitor 31 and a grid parasitic suppression resistor 39.Cathode self-bias is provided by resistor 4l in the cathode return.Conventional screen bias for all the screen grid tubes is obtained froma source of positive voltage applied to terminal 43. The plate loadincludes a resonant circuit or band pass filter 44 tuned to 1000 cyclesand comprising a 1.0 henry choke 45 and a 0.025/ifd capacitor 41, thehigh potential ends of the capacitor and choke being connected to platethrough damping resistors 5| and 53, respectively. The common lowpotential end of the filter is by-passed to cathode and connected toterminal 43 through an isolating resistor 49. The plate of the firstamplifier tube 35 is connected to the grid of the second amplifier tube55 through a high pass filter 46 comprising parallel-connected capacitor51, resistor 59, and the resistor 63. The midpoint of the two resistorsis connected to the grid of tube 55 through a parasitic reducingresistor 5|. Since there is a D. C. path between the second amplifiergrid and the first amplifier plate, the grid must be held at a stable,relatively low D. C. potential. This is accomplished by connecting theground end of resistor 63 to a source of negative D. C. potentialderived from a resistor network including a potentiometer 65 and twofixed resistors 61 and 69 so connected across a negative potentialsource available at terminal 1I that the effective grid voltage may bemade approximately zero.

The second amplifier is connected similarly, except that a low passcoupling network 48 is used, including series connected capacitor 13 andresistor 15, both in shunt with another resistor 11. Effectively zerogrid bias for the third amplifier tube 16 is provided through resistor18 by potentiometer 19 connected to the same source of negativepotential available at terminal 1|. The plate of the second amplifier 55is coupled to the third amplifier grid through a second high pass filter50 comprising resistor 18 and the shunt connected resistor 8| andcapacitor 83. A small parasitic suppression resistor may also be used.

In order to provide output voltage at a suflicient level and impedanceto feed a large number of parallel potentiometers, two dual triodeampliers B1 and 89 of the 6J6 type are employed. ,All the grids areconnected in parallel, and coupled to the plate of the third amplifier16 through a large coupling capacitor 9| and a high pass filter network52 including shunt-connected resistor 93 and capacitor 95, both inseries with a grid resistor |03. The plate load of amplifier 15 includesa low pass network 54 comprising resistor 91 in parallel with seriesconnected resistor 99 and capacitor I0|. Grid bias for the output tubesis provided through grid resistor |03, the lower end of which isconnected to a voltage divider comprising resistors |05 and |01connected between ground and terminal 43. The cathode electrodes of theoutput tubes are all connected together and to an output terminal |09.The plate electrodes are all connected together through parasiticsuppression resistors to terminal Low pass filter networks ||3 and ||5are connected between the output terminals and ground. The plate andcathode circuits to B+ and ground, respectively, are completed throughthe Micropots in the A networks ||1, as indicated above.

To provide the desired degree of accuracy the overall amplifier gain at1000 cycles has been made approximately 15,000. High gain amplifiers,however, may become unstable particularly where there is a directfeedback path from output to input. Self oscillation or otherinstability will prevent solution of the equation. The conditions forstability have been stated above, namely that the 1000 cycle phase shiftshall be 0 and 180 and that the added phase shift or phase margin forother frequencies shall not exceed 190 except for ,frequencies so farremoved from the operating frequency that the net amplifier gain atthose frequencies is less than unity. Such a design insures stabilityfor any degree of feedback required to represent a system ofsimultaneous equations whose matrix has no characteristic roots with aphase angle greater than i. Within this range are all passive linearelectrical network problems, that is, networks which do not includevoltage generators or negative resistance. However, any system ofsimultaneous linear equations may be solved, provided the solution isfinite, if a transformation is made to convert the system into one whichmeets the above limitation before applying the equation to theelectrical equation solver. This conversion is well known by thoseskilled in the art and need not be explained further here.

In Fig. 4, the actual values of the components 9 necessary to meerl theabove requirement have been shown. However, it is not intended that thespecific circuit and specific values illustrated shall be a limitation,since those skilled in the art will recognize modifications which may bemade to accomplish the same result.

With regard to the amplification of the normal 1000 cycle signal, theoperation of the amplifier is straightforward. The only care that mustbe exercised is to insure thatI the reactive elements are chosen so asto produce a .negligible ,phase shift in the coupling circuits so thatthe overall phase shift will be or 180. plished, for example, byutilizing resonant circuits or by selecting series-connected capacitorsThis is accomof negligible impedance at 1000 cycles, and paralmet overthe range of frequencies which it is necessary to consider.

Considering first the resonant circuit M ln the plate return of tube 35,illustrated separately in Fig. '7, inductive and capacitive reactances Land C were selected which resonated at 1000 cycles. It is known that atresonance the phase shift is zero, and that on either side of resonancethe phase shift approaches 90 and is positive in one direction andnegative in the other. The effect of resistors R (I and 53 in Fig. 4) isto limit the maximum phase shift to a value substantially less than 90.The actual frequency vs. phase shift curve of the circuit employed risesrapidly to a maximum of '14 at 2000 cycles and 500 cycles and thengradually reduces to a few degrees at 100 kc. and 10 cycles,respectively. The low frequency portion of this curve is shown as curveA of Fig. 8. The high frequency portion of the curve is shown at A' inFig. 9.

'The high pass filters are each of the type illustrated in Fig. 5 inwhich It will be noted that for very low frequencies C can be neglectedand the coupling circuit is then simply a voltage divider with a stepdown ratio of :1. At high frequencies, however, C effectivelyshort-circuits R1, and the full output is applied to the succeedinggrid. The added phase shift is negligible at relatively low andrelatively high frequencies, the actual frequencies where this occursbeing determined by the size of C, and is a maximum at the intermediatefrequency at which the capacitive reactance Xe equals 2.85Rz. However,as is well known, where the ultimate attenuation is 10:1, i. e., wherethe maximum phase shift does not exceed 55. In order to keep th'e netadded phase shift from exceeding 90 it is therefore necessary to staggerthe points of maximum phase shift of the different high pass filterswith respect to frequency of occurrence, as shown in Fig. 8.

l0 Since the phase shift for frequencies below 1000 cycles due to theresonant plate load was known to reach a relatively low value, say lessthan 40, at a frequency of 85 cycles, (see curve A, Fig. 8) the curve Bof the nearest high pass network should be so located that the phaseshift y for frequencies above, say 75 cycles, does not exceed about thesame value. The resultant (curve R, Fig. 8) will then be less than 90withsome margin of safety. The point on the relative frequency vs. phaseshift curve ofl a high pass lterof the type illustrated in Fig. 5 'atwhich the phase shift equals 40 is the point at which the capacitivereactance equals' the Value of R2. Consideringflrst the filter 52 in theoutput of tube 16, R2 (resistor |03) is seen to be .91 megohm. Thecapacitor whose reactance at 75 cycles is equal to .91 megohm is one ofapproximately 2340 auf. This is therefore the value selected forcapacitor 95.

The high pass lter 50 in the output of tube 55 was then determined inthe same manner. A

, capacitor of .0281 nf. has .91 megohm reactance .30 that the phasemargin requirements have been at 6.25 cycles. A value of .0281 afd. wastherefore selected for capacitor 83. The phase shift is as shown bycurve C of Fig. 8. The value of capacitor 51 was then determined in thesame manner, and the phase shift is as shown by curve D, Fig. 8. It isnot necessary to consider any lower frequencies, since the amplifiergain has reached a value less than unity at a frequency of 0.19 cycle,as shown by Fig. 11, and consequently oscillation cannot take place forany value of feedback.

Referring toFig. 11, the attenuation of each of the high pass filtershas been plotted separatelyand the resultant attenuation shown. Curve Ais the attenuation of high pass filter 52, B the attenuation of filter50, C that of filter 46 and D that of the band pass filter 44, all shownfor the range below 1000 cycles and plotted in db. Curve R is theresultant attenuation considering all filters. Since an attenuation of83.5 db. corresponds to an attenuation of 15,00021, and since theamplifier gain is 15,000, it follows that at 0.19 cycle, and below, theresultant gain of the amplifier is less than unity. This is the lowerfrequency of the amplifier employed which need be considered indetermining the resultant phase margin. Fig. 8 shows that the phaseshift does not exceed throughout this range, and the required operatingcharacteristic for frequencies below'1000 cycles has therefore been met.

The low pass filters are designed similarly. Fig. 6 represents thecircuit utilized.

is made equal to 9, as before, so that the maximum phase shift does notexceed 55. The point at which the phase shaft reaches approximately 40occurs when X=Ri. This point is then used to calculate the values of thecapacitors. Fig. 9, curve A' shows the phase shift above 1000 cycles dueto the resonant circuit 44 in the output of tube 35. 11 kc. was taken asthe point at which the low pass filter 54 associated with tube 16 wouldcross the first curve, as shown by curve B'. R1 (resistor 91) is 5100 w.The capacity whose reactance at 13 kc, is 5100 (necessary to make thetwo curves cross at 11 kc.) is approximately 2400 upf., the valueselected for capacitor 10|. It can be seen, therefore, that capacitor 13in the output circuit of tube 55 will be 200 paf., and the phase shiftas represented by curve C' of Fig, 9.

l l In determining the-'value for the capacitor in filters ||3 and ||5,the value of R1 is determined by the parallel impedance of n A"networks. In

, the case when n=10, and Athe total resistance of each Micropot is 1000w, Ithe eii'ective value of R1, considering the shunt drive impedance ofthe driver tu-bes, (cathode circuit) is 33 w. The capacitor whosereactance at 1.7 mc. is 33 w is 2800 auf. A standard value of 2400 aufwas selected. The phase shift is as illustrated by curve D', Fig. 9. Theresultant added phase shift is shown by curve R', Fig. 9, and it is seenthat it does not exceed 90. Values above 5400 kc. need not be consideredsince the amplifier gain is then less than unity, as will appear fromFig. 12 to which reference is now made.

The individual attenuation curves of the low pass filters and the bandpass filter are plotted against frequency in the range above 1000cycles. Curve D is the band pass lter 44, E represents Ithe attenuationof the illters I I3 or I5, F the illter 54, and G the filter 48. Theresultant R represents the effective attenuation above 1000 cycles. Itwill beV seen that an attenuation of 15,000:1 (83.5 db.)`is reached at5.4 megacycles. This frequency is therefore the upper limit of the rangeover which t'he phase shift need be considered. Fig. 9 shows that theoverall phase shift is less .than 90 throughout this range, and thedesign requirement has therefore been met for the high frequency rangealso.

Referring now to Fig. 10, the components of the signal generator, bridgeand indicator |20 are shown in schematic form. The 1000 cycle voltagewhich is used to actuate the Y networks of Fig. 2 is obtained from aconventional 1000 cycle oscillator |35, -the output of which is appliedthrough a gain control potentiometer |31 to the input of a conventionaldriver amplifier |39. The amplifier output is applied to the fourparallel connected grid electrodes of two power amplier tubes of the 6J6type, |4| and |43. Parasitic limiting grid resistors |45 may be includedin each of the grid leads. The four plate electrodes are also connectedin parallel through low resistance parasitic limiting resistors |41 andthence to the output plate terminal |49 which is the terminal P shown inFig. 2. The two` cathode electrodes of the output tubes are connectedtogether to the output terminal |5| which 'is the K terminal shown inFig. 2.

The power amplifier output, that is, the plate and cathode terminals |49and |5I, are also connected to a four pole double throw polarityreversing switch |53 which functions in the same manner as the polarityreversing switches shown in Fig. 2, to connect a Micropot |55 in theplate circuit and a resistor |51, having the same value of resistance,in the cathode circuit when the switch is in one position, and reversingthe position of the Micropot and resistor in the other position of theswitch. As before, the B (or ground) and +B connections to a source ofpotential of the order of 150 volts direct current, not shown, are madeby the polari l reversing switch.

The movable contact of th Micropot |55 applies a 1000 cycle voltage, ofone phase or the other, to a. bridge circuit which includes in the ordernamed, a phasing network |61, -a balancing potentiometer |59, a secondphasing network |1| and a voltage divider which includes a resistornetwork comprising a first resistor |13 and a second resistor |15. 'I'heratio of the two resistors is selected so that a voltage division of 10to 1 is produced when connection is made to their midpoint as b v meansof a multiplier switch '|11. The lower end of resistor |15 is groundedand the opposite end of resistor |13 connected 4to the movable arm of amultipoint contact switch 19. Various positions of the selector switch|19 are connected to input terminals X1, X2, Xn which are the similarlyidentified terminals shown in Fig. 2 to which the unknown X voltages areconnected. Inl addition two positions are provided for the plate P andcathode K terminals of the power amplier for the purpose of calibration.Resistor |13 of the multiplier network may be shunted by a smallvariable capacitor for balancing the phase of the X voltage applied tothe bridge when the switch |11 is in the multiplyby-10 position.

The movable arm of potentiometer |69 is coupled through capacitor |8|-to gain control potentiometer |83 which applies the voltage developedby the bridge to the input of a 1000 cycle ampliiler |85. The output ofthis amplier is connected to the vertical defiecting electrodes of aconventional cathode ray indicator tube |81. Synchronized horizontalscanning voltage for Athe cathode ray tube is applied by a suitableampliiier |89, the input of which is taken from the output of theoscillator |35.

'Ihe system is adjusted in the following manner. Place multiplier switch|11 on the 1:1 position. Place selector switch |19 on position P. Placepolarity switch |53 in the -i position so that the Micropot is in thecathode circuit. Turn the Micropot control to a maximum output. Voltagesof equal amplitude and opposite phase are then being impressed acrossthe bridge. If the cathode ray trace is elliptical, adjust the phasingcapacitors in the bridge circuits to balance the phase and producestraight line trace. Adjust potentiometer |59 to null position indicatedby absence of vertical deflection. Then place selector switch |19 onposition K, reverse the position of switch |53 and recheck.

The bridge and indicator are of the null type. Having set the various Y"networks of Fig. 2

in accordance with the similarly identified con stants of thesimultaneous equations to be solved and also set the necessary number ofA" networks to values representing the coefiicients of X1. X2, Xn, theactual value X1 is rst determined Iby placing selector switch |19 to theX1 position, This applies an alternating Voltage of a certain amplitudeacross the multiplier network. A voltage of the same frequency is alsopresent at the terminal of the Micropot |55. Switch |53 is operated tomake these voltages of opposite phase. If they are not of equalamplitude, a voltage will appear at the input of amplifier |85 and avertical deilection will be produced in the cathode ray tube producing aline at an angle which depends on their relative amplitudes. AdjustingMicropot |55 will then vary the amplitude of the reference voltage. Thisis done until the two voltages are exactly equal in amplitude at whichpoint there will be no vertical defiecting voltage applied to thecathode ray tube and only a single horizontal line will be produced. Atthis point of balance the reference voltage developed by Micropot |55 isexactly equal to the unknown voltage X1, and its amplitude may bemeasured by any suitable voltmeter or, preferably, determined from thecalibration of the Micropot, always lconsidering the position of the10:1 voltage divider switch |11.

It will be observed that with the bridge system illustrated, the actualamplitude of the reference voltage need not be determined, since thesame voltage is applied both to the bridge input and to the "Y networks,and any change in the amplitude of this voltage is compensated by reasonof the fact that a like change is applied both to the balancing voltageand to the input of the networks. The relative Value is determined,however, by the calibration of the Micropot |55 and this indicates thevalue of the unknown Xi. Its sign is determined by the position ofpolarity reversing switch |53. If the X voltage is of suiiicientamplitude that a balance cannot be obtained within the range of Micropot|55, multiplier switch |11 is used to reduce the X voltage in the ratioof 1:10. The indicated value of X1 is then multiplied by 10 to obtainthe proper value. The values of Xn, 2Q. Xn are then determined insequence in the same manner.

It will be appreciated that any equation may be multiplied or divided byany number without affecting its identity. For this reason it is weil toconsider the range of the coefficients and constants in the equations tobe solved, and multiply or divide one or more equations by 10 or a powerthereof, as the case may be, to bring all equations within the samegeneral range. This is necessary in order to take advantage ofthemaximum possible accuracy. The potentiometer dials are of the countertype, in which the dial is rotated 10 times to complete the movement ofthe contact arm 'from one extreme to the other. A counter indicates howmany revolutions have been made. Reading first the counter and then thedial calibration, which reads from to 100, the numbers 763, for example,may be multiplied or divided by powers of as desired to suit the actualvalues of the constants and coeiiicients of the equation in question.However, it is important that for any equation the same range be usedthroughout. Thus, maximum accuracy is achieved for the equation when thecalibration maximum is assumed to be 10, since this equation can be setto the following accuracy: 3.00, 4.00 and 7.00.

What we claim is:

l. A simultaneous equation solver including means providing a firstgroup of constant frequency voltages each of which has an amplitudeproportional to the value of the constant term of a different one ofsaid equations and has a phase of 0,0r 180 electrical degrees dependingon the signof said value, separate amplifiers each having applied to itsinput circuit a different one of the voltages of said first group, amatrix connected to the output circuits of said amplifiers for providingother voltages representative of the product of each coeflicient and itsassociated unknown in said equations, means connected between saidmatrix and the inputs of said amplifiers for combining with eachconstant term representing voltage the ones of said other voltages whichrepresent the product of each coeiiicent and its associate unknown inthe equation containing that constant, and filter means interconnectedwith each of said amplifiers to maintain the phase of the input voltageof said amplifiers at 0 or 180 electrical degrees for said constantfrequency and within a phase margin of i 90 electrical degrees for otherfrequencies at which the gain of said amplifier is greater than unity.

2. A simultaneous equation solver including means providing a firstgroup of constant frequency voltages each of which has an amplitudeproportional to the value of the constant term of a different one ofsaid equations and has a phase of 0 or 180 electrical degrees dependingon the sign of said value, separate amplifiers each having applied toits input circuit a different one of the voltages of said first group, amatrix connected to the output circuits of said amplifiers for providingother voltages representative of 'che product of each coefficient andits associated unknown in said equations, means connected between saidmatrix and the inputs of said amplfiers for combining with each constantterm representing voltage the ones of said other voltages whichrepresent the product of each coefficient and its associate unknown inthe equation containing that constant, filter means interconnected witheach of said amplifiers to maintain the phase of vthe input voltage ofsaid amplifiers to maintain the phase of the input voltage of saidamplifiers at 0 or 180 electrical degrees for said constant frequencyand within a phase margin of i electrical degrees for other frequenciesat which the gain of said amplifiers is greater than unity, and meansincluding a bridge circuit connected to provide a resultant voltageindicative of the values of the unknown terms of said equations.

3. A simultaneous equation solver including means providing a firstgroup of constant frequency voltages each of which has an amplitudeproportional to the value of the constant term of a diierent one of saidequations and has a phase of 0 or 180 electrical degrees depending onthe sign of said value, separate amplifiers each having applied to itsinput circuit a diierent one of the voltages of said iirst group, amatrix connected to the output circuits of said amplifiers for providingother voltages representative of the product of each coefficient and itsassociated unknown in said equations, means connected between saidmatrix and the inputs of said amplifiers for combining with eachconstant term representing voltage the ones of said other voltages whichrepresent the product of each coefficient and its associate unknown inthe equation containing that constant, each of said amplifiers havingits different stages intercoupied through filter circuits which havetheir points of maximum phase shift staggered so that the phase of theinput voltage of said amplifiers is maintained within a phase margin ofi90 electrical degrees at frequencies which dler from said constantfrequency and have their amplitude increased by said ampliers.

GEORGE W. BROWN. EDWIN A. GOLDBERG.

No references cited.

